How do you create a curved sequence of vertices?
Since you are using a mesh object, select the single vertex and use Ctrl + B to start the bevel, and in the Operator Panel (F6 for popup) tick the option 'Vertex Only' and then increase the number of Segments to get a nice curve. 
Here I have completed this process on a simple plane with the face deleted so as to imitate your scenario.

After, you can turn on Proportional Edit with 'O' and use the MMB to increase/decrease influence and move the vertices around to shape them.